1. A triangulation of a polygon is a way to divide the polygon into triangles by adding diagonals that do not
intersect each other (except possibly at the vertices of the polygon). How many diagonals does one need to add
in order to triangulate a regular 2021-gon?
2. Let a, b, c, d, and e be five consecutive terms in an arithmetic sequence, and suppose that a + b + c + d + e = 40.
Which of a, b, c, d, or e can be found? (ITS NOT 6+7+8+9+10, it can only be one of the letters.)
3.A chef places four tablespoons of water into an eight-tablespoon measuring cup and four tablespoons of vinegar
into a second cup of the same size. She then pours half the water from the first cup to the second and, after
stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first
cup is now vinegar?
1. From the formula, a regular 2021-gon has 2021*2018/2 = 2039189 diagonals.
2. You can find a, becuase a is part of the formula for the sum
a + (a + d) + (a + 2d) + ... + (a + (n - 1)d) = (2a + (n - 1)d)/2.
+128460 2. Let a, b, c, d, and e be five consecutive terms in an arithmetic sequence, and suppose that a + b + c + d + e = 40.
Which of a, b, c, d, or e can be found?
c, the middle term, is the one that can be found
We have an odd number of terms..so this will = the sum of the terms / the number of terms = 40 / 5 = 8
To see this...suppose we have
1 + 4 + 7 + 10 + 13 = 35
The middle term is 35/5 = 7
+128460 3.A chef places four tablespoons of water into an eight-tablespoon measuring cup and four tablespoons of vinegar
into a second cup of the same size. She then pours half the water from the first cup to the second and, after
stirring thoroughly, pours half the liquid in the second cup back to the first. What fraction of the liquid in the first
cup is now vinegar?
After pouring, the first cup contains 2 Tsp of water
And the mixture in the second cup is 2 parts water and 4 parts vinegar = 2 Tsp of water and 4 Tsp of vinegar
The mxing does not affect the ratio
So..... she pours 1/2 of this back into the first cup = 1 Tsp of water and 2 Tsp of vinegar
And the first cup now contains 3 Tsp of water and 2 Tsp of vinegar
So....the first cup contains 2 / ( 2 + 3) = 2/5 vinegar
